What type of a quadrilateral is this?

Question

Homi · Accepted Answer

We need to identify the type of quadrilateral shown in the diagram.

Step 1 — Identify parallel lines and transversal

The diagram shows a quadrilateral where the top side (line segment AD') is parallel to the bottom side (line segment DA'). A line segment BC acts as a transversal, cutting these two parallel lines.

Step 2 — Analyze the given angles

The angles formed by the transversal BC with the parallel lines are given. At point B, the interior angle $\angle ABC$ is shown as the sum of two smaller angles, $x$ and $y$ . So, we can write: $\angle ABC = x + y$ At point C, the interior angle $\angle BCD$ is also shown as the sum of two smaller angles, $y$ and $x$ . So, we can write: $\angle BCD = y + x$ This means that the two interior angles on the same side of the transversal are equal. $\angle ABC = \angle BCD$

Step 3 — Apply properties of parallel lines

When two parallel lines are cut by a transversal, the sum of the consecutive interior angles (angles on the same side of the transversal) is $180^\circ$ . In our case, the parallel lines are AD' and DA', and the transversal is BC. So, the sum of $\angle ABC$ and $\angle BCD$ must be $180^\circ$ . $\angle ABC + \angle BCD = 180^\circ$ We already found that $\angle ABC = \angle BCD$ . Let's substitute $\angle BCD$ with $\angle ABC$ in the equation. $\angle ABC + \angle ABC = 180^\circ$ $2 \times \angle ABC = 180^\circ$ Now, we can find the value of $\angle ABC$ . $\angle ABC = \frac{180^\circ}{2}$ $\angle ABC = 90^\circ$ Since $\angle ABC = \angle BCD$ , we also have: $\angle BCD = 90^\circ$ This means that the transversal line segment BC is perpendicular to both parallel lines AD' and DA'.

Step 4 — Determine the type of quadrilateral

The quadrilateral shown is $ADD'A'$ . We know that the side AD' is parallel to the side DA'. This means the quadrilateral is a trapezoid. The diagram also shows that the non-parallel sides AD and D'A' are parallel to each other. This is indicated by the visual representation of the figure, which is a common way to draw a parallelogram. If both pairs of opposite sides are parallel, the quadrilateral is a parallelogram. We have found that the transversal BC is perpendicular to the parallel sides AD' and DA'. This means the height of the parallelogram is equal to the length of BC. However, this information alone does not tell us if the angles of the parallelogram (like $\angle D$ , $\angle A$ , $\angle D'$ , $\angle A'$ ) are $90^\circ$ . A parallelogram with one right angle is a rectangle. But we don't have information about the angles of the parallelogram itself. Based on the visual representation and the deduction that it has two pairs of parallel sides, the most specific type of quadrilateral we can identify is a parallelogram.

Answer

(i) The quadrilateral is a parallelogram.

Diagram 1

Question 26

Answer

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